Formula Sheets Shear Force & Bending Moment Diagrams (SFD & BMD) - Solid

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Shear F orce and Bending Momen t Diagrams
F orm ula Sheet for Mec hanical G A TE
Shear F orce
• Shear F orce (V ) : In ternal force resisting transv erse loading.
V =
?
F
y
where F
y
are v ertical forces (N ) to the left of the section.
• Relation with Distributed Load : Shear force is the in tegral of the distributed
load.
dV
dx
= -w(x)
where w(x) is the distributed load in tensit y (Nm
-1
).
• Change in Shear F orce : F or a distributed load o v er length ?x .
?V = -
?
w(x)dx
• P oin t Load Effect : A t a p oin t load P , shear force c hanges abruptly .
V
after
= V
b efore
-P
Bending Momen t
• Bending Momen t (M ) : In ternal momen t resisting rotational loading.
M =
?
(F
y
·d)
where d is the p erp endicular distanc e from the section to the force (Nm ).
• Relation with Shear F orce : Bending momen t is the in tegral of shear force.
dM
dx
= V
• Change in Bending Momen t : F or a shear force o v er length ?x .
?M =
?
V(x)dx
• P oin t Momen t Effect : A t an applied momen tM
0
, b ending momen t c hanges abruptly .
M
after
= M
b efore
+M
0
1
Page 2


Shear F orce and Bending Momen t Diagrams
F orm ula Sheet for Mec hanical G A TE
Shear F orce
• Shear F orce (V ) : In ternal force resisting transv erse loading.
V =
?
F
y
where F
y
are v ertical forces (N ) to the left of the section.
• Relation with Distributed Load : Shear force is the in tegral of the distributed
load.
dV
dx
= -w(x)
where w(x) is the distributed load in tensit y (Nm
-1
).
• Change in Shear F orce : F or a distributed load o v er length ?x .
?V = -
?
w(x)dx
• P oin t Load Effect : A t a p oin t load P , shear force c hanges abruptly .
V
after
= V
b efore
-P
Bending Momen t
• Bending Momen t (M ) : In ternal momen t resisting rotational loading.
M =
?
(F
y
·d)
where d is the p erp endicular distanc e from the section to the force (Nm ).
• Relation with Shear F orce : Bending momen t is the in tegral of shear force.
dM
dx
= V
• Change in Bending Momen t : F or a shear force o v er length ?x .
?M =
?
V(x)dx
• P oin t Momen t Effect : A t an applied momen tM
0
, b ending momen t c hanges abruptly .
M
after
= M
b efore
+M
0
1
Shear F orce and Bending Momen t Diagrams
• Shear F orce Diagram (SFD) :
– Plot V vs. x .
– Constan t w(x) : Linear V.
– Linear w(x) : P arab olic V.
– P oin t load: Step c hange in V.
• Bending Momen t Diagram (BMD) :
– Plot M vs. x .
– Constan t V : Linear M.
– Linear V : P arab olic M.
– P oin t momen t: Step c hange in M.
• Sign Con v en tion :
– P ositiv e V : Up w ard force on the left or do wn w ard on the righ t.
– P ositiv e M : Causes compression in the top fib ers, tension in the b ottom.
Stress Due to Bending
• Flexural F orm ula : Normal stress due to b ending.
s=
My
I
where M is the b ending momen t, y is the distance from the neutral axis (m ), I is the
momen t of inertia (m
4
).
• Maxim um Bending Stress : Occurs at the outermost fib er.
s
max
=
Mc
I
where c is the distance from the neutral axis to the outermost fib er.
• Shear Stress in Beams : Shear stress due to shear force.
t =
VQ
Ib
where Q is the first momen t of area ab o v e the p oin t ( m
3
), b is the width of the b eam
(m ).
Key Relationships
• Maxim um/Minim um Bending Momen t : Occurs where V = 0 or at p oin t mo-
men ts.
2
Page 3


Shear F orce and Bending Momen t Diagrams
F orm ula Sheet for Mec hanical G A TE
Shear F orce
• Shear F orce (V ) : In ternal force resisting transv erse loading.
V =
?
F
y
where F
y
are v ertical forces (N ) to the left of the section.
• Relation with Distributed Load : Shear force is the in tegral of the distributed
load.
dV
dx
= -w(x)
where w(x) is the distributed load in tensit y (Nm
-1
).
• Change in Shear F orce : F or a distributed load o v er length ?x .
?V = -
?
w(x)dx
• P oin t Load Effect : A t a p oin t load P , shear force c hanges abruptly .
V
after
= V
b efore
-P
Bending Momen t
• Bending Momen t (M ) : In ternal momen t resisting rotational loading.
M =
?
(F
y
·d)
where d is the p erp endicular distanc e from the section to the force (Nm ).
• Relation with Shear F orce : Bending momen t is the in tegral of shear force.
dM
dx
= V
• Change in Bending Momen t : F or a shear force o v er length ?x .
?M =
?
V(x)dx
• P oin t Momen t Effect : A t an applied momen tM
0
, b ending momen t c hanges abruptly .
M
after
= M
b efore
+M
0
1
Shear F orce and Bending Momen t Diagrams
• Shear F orce Diagram (SFD) :
– Plot V vs. x .
– Constan t w(x) : Linear V.
– Linear w(x) : P arab olic V.
– P oin t load: Step c hange in V.
• Bending Momen t Diagram (BMD) :
– Plot M vs. x .
– Constan t V : Linear M.
– Linear V : P arab olic M.
– P oin t momen t: Step c hange in M.
• Sign Con v en tion :
– P ositiv e V : Up w ard force on the left or do wn w ard on the righ t.
– P ositiv e M : Causes compression in the top fib ers, tension in the b ottom.
Stress Due to Bending
• Flexural F orm ula : Normal stress due to b ending.
s=
My
I
where M is the b ending momen t, y is the distance from the neutral axis (m ), I is the
momen t of inertia (m
4
).
• Maxim um Bending Stress : Occurs at the outermost fib er.
s
max
=
Mc
I
where c is the distance from the neutral axis to the outermost fib er.
• Shear Stress in Beams : Shear stress due to shear force.
t =
VQ
Ib
where Q is the first momen t of area ab o v e the p oin t ( m
3
), b is the width of the b eam
(m ).
Key Relationships
• Maxim um/Minim um Bending Momen t : Occurs where V = 0 or at p oin t mo-
men ts.
2
• P oin t of Con traflexure : Where M =0 (b ending momen t c hanges sign).
• Load, Shear, and Momen t Relationship:
w(x)= -
dV
dx
, V =
dM
dx
Key Notes
• Use SI units: V in N , M in Nm , w in Nm
-1
.
• Dra w FBD (F ree Bo dy Diagram) to calculate reactions b efore constructing SFD and
BMD.
• F or distributed loads, compute the equiv alen t p oin t load and its lo cation for reactions.
• Chec k b oundary conditions (e.g., V =0 , M =0 at free ends; M =0 at hinges).
• Maxim um shear stress in rectangular b eams o ccurs at the neutral axis.
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